20 seconds of dimensional analysis are more than sufficient to disprove the above formula.
Areas must have units equal to length squared, but this formula has units equal to length, not length squared.
For instance, if A and B and C are in inches, the S is also in inches, so it cannot be an area. It might be some sort of formula for perimeter, or semi-perimeter, or harmonic mean of the sides, or some other metric that has a dimension of length, but definitely not area.
It took longer to write this down than to it took to figure out.
And, by the way, that holds for any ABC, not just fibonacci numbers. So I have proved (or disproved) more than requested. Extra credit, Ady?